This is the first systematic study exploring the potential of high-energy EXAFS as a structural tool for lanthanoids and third-row transition elements. The K-edge X-ray absorption spectra of the hydrated lanthanoid(III) ions both in aqueous solution and in solid trifluoromethanesulfonate salts have been studied. The K-edges of lanthanoids cover the energy range from 38 (La) to 65 keV (Lu), while the corresponding energy range for the L3-edges is 5.5 (La) to 9.2 keV (Lu). We show that the large widths of the core-hole states do not appreciably reduce the potential structural information in the high-energy K-edge EXAFS data.Moreover, for lanthanoid compounds, more accurate structural parameters are obtained from analysis of K-edge than from L3-edge EXAFS data. The main reasons are the much wider k range available and the absence of double-electron transitions, especially for the lighter lanthanoids. A comparative K- and L3-edge EXAFS data analysis of nonahydrated crystalline neodymium(III) trifluoromethanesulfonate demonstrates the clear advantages of K-edge analysis over conventionally performed studies at the L3-absorption edge for structural investigations of lanthanoid and third-row transition metal compounds. The coordination chemistry of the hydrated lanthanoid(III) ions in aqueous solution and solid trifluoromethanesulfonate salts, based on the results of both the K- and L3-edge EXAFS data, is thoroughly discussed in the next paper in this series